arcsinx=x+ 1/6x^3+o(x^3)

tanx=x+ 1/3x^3+o(x^3)

arctanx=x- 1/3x^3+o(x^3)

ln( 1+x)=x- 1/2x^2+o(x^2)

cosx= 1- 1/2x^2+o(x^2)

The above applies to Taylor expansion when x tends to 0.

Extended data:

Taylor formula can express a function by adding several terms, and these added terms are obtained from the derivative of the function at a certain point.

In mathematics, Taylorseries represents a function by a series of infinite terms, and these added terms are obtained by the derivative of the function at a certain point. Taylor series is named after British mathematician SirBrookTaylor, who published Taylor formula in 17 15.

Taylor series derived from the derivative of a function at the zero point of the independent variable is also called McLaughlin series, which is named after the Scottish mathematician colin maclaurin. Taylor series plays an important role in approximate calculation.

Definition: If there is any derivative at x=x0, the power series is called Taylor series at x0. ?

In Taylor's formula, take x0=0 and get the series.

The importance of Taylor series is reflected in the following three aspects:

The derivation and integration of 1 power series can be carried out item by item, so the summation function is easier.

Through analytic continuation, the analytic function can be extended to Taylor series defined in the open domain on the complex plane, thus making complex analysis possible.

Taylor series can be used to approximately calculate the function value.

For some infinitely differentiable functions, although their expansions converge, they are not equal to f(x). For example, for piecewise function, when x≠0 and f(0)=0, all derivatives are zero when x=0, so the Taylor series of this f(x) is zero and the convergence radius is infinite, although this function f is zero only when x=0. However, this problem does not hold in the complex variable function, because z tends to zero along the imaginary axis and does not tend to zero.

Some functions cannot be expanded into Taylor series because there are some singularities. But if the variable x is a negative exponential power, we can still expand it into a series. For example, it can be expanded into Laurent series.

Basic principle: the k-fold irreducible factor of polynomial is the k- 1 fold irreducible factor of its WeChat service;

Basic idea: Using Wechat quotient coefficient to study the properties of arbitrary function by polynomial (mainly convergence in undergraduate course).

References:

Baidu Encyclopedia-Taylor Series